Measurable multiresolution systems, endomorphisms, and representations of Cuntz relations. (English) Zbl 07886851
Summary: The purpose of this paper is to present new classes of function systems as part of multiresolution analyses. Our approach is representation theoretic, and it makes use of generalized multiresolution function systems (MRSs). It further entails new ideas from measurable endomorphisms dynamics. Our results yield applications that are not amenable to more traditional techniques used on metric spaces. As the main tool in our approach, we make precise new classes of generalized MRSs which arise directly from a dynamical theory approach to the study of surjective endomorphisms on measure spaces. In particular, we give the necessary and sufficient conditions for a family of functions to define generators of Cuntz relations. We find an explicit description of the set of generalized wavelet filters. Our results are motivated in part by analyses of sub-band filters in signal/image processing. But our paper goes further, and it applies to such wider contexts as measurable dynamical systems and complex dynamics. A unifying theme in our results is a new analysis of endomorphisms in general measure space, and its connection to multi-resolutions, to representation theory, and generalized wavelet systems.
MSC:
| 81-XX | Quantum theory |
| 46G12 | Measures and integration on abstract linear spaces |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 20K30 | Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups |
| 22E66 | Analysis on and representations of infinite-dimensional Lie groups |
Keywords:
measure spaces; endomorphisms; multiresolution; wavelet filters; Cuntz relations; Radon-Nikodym derivative; Markovian functionsReferences:
| [1] | Alpay, D.; Colombo, F.; Sabadini, I.; Schneider, B., Beurling-Lax type theorems and Cuntz relations, Linear Algebra Appl., 633, 152-212, 2022 · Zbl 07432147 · doi:10.1016/j.laa.2021.10.008 |
| [2] | Alpay, D.; Jorgensen, P.; Lewkowicz, I., Characterizations of families of rectangular, finite impulse response, para-unitary systems, J. Appl. Math. Comput., 54, 1-2, 395-423, 2017 · Zbl 1409.93022 · doi:10.1007/s12190-016-1015-x |
| [3] | Alpay, D., Jorgensen, P., Lewkowicz, I.: \(W\)-Markov measures, transfer operators, wavelets and multiresolutions. In: Frames and harmonic analysis, Contemp. Math., vol. 706, pp. 293-343. Amer. Math. Soc., Providence (2018) · Zbl 1398.37020 |
| [4] | Alpay, D.; Jorgensen, P.; Lewkowicz, I., Representation theory and multilevel filters, J. Appl. Math. Comput., 69, 2, 1599-1657, 2023 · Zbl 1539.42040 · doi:10.1007/s12190-022-01805-z |
| [5] | Andrianov, PA, Multidimensional periodic discrete wavelets, Int. J. Wavelets Multiresolut. Inf. Process., 20, 2, 2150053, 2022 · Zbl 1490.42038 · doi:10.1142/S0219691321500533 |
| [6] | Baggett, LW; Larsen, NS; Packer, JA; Raeburn, I.; Ramsay, A., Direct limits, multiresolution analyses, and wavelets, J. Funct. Anal., 258, 8, 2714-2738, 2010 · Zbl 1202.42060 · doi:10.1016/j.jfa.2009.08.011 |
| [7] | Baggett, LW; Merrill, KD; Packer, JA; Ramsay, AB, Probability measures on solenoids corresponding to fractal wavelets, Trans. Am. Math. Soc., 364, 5, 2723-2748, 2012 · Zbl 1246.42032 · doi:10.1090/S0002-9947-2012-05584-X |
| [8] | Bénéteau, C., A natural extension of a nonsingular endomorphism of a measure space, Rocky Mt. J. Math., 26, 4, 1261-1273, 1996 · Zbl 0883.28016 · doi:10.1216/rmjm/1181071987 |
| [9] | Bezuglyi, S., Jorgensen, P.E.T.: Representations of Cuntz-Krieger relations, dynamics on Bratteli diagrams, and path-space measures. In: Trends in Harmonic Analysis and Its Applications, Contemp. Math., vol. 650, pp. 57-88. Amer. Math. Soc., Providence (2015) · Zbl 1351.46064 |
| [10] | Bezuglyi, S., Jorgensen, P.E.T.: Transfer operators, endomorphisms, and measurable partitions, Lecture Notes in Mathematics, vol. 2217. Springer, Cham (2018) · Zbl 1416.37002 |
| [11] | Bhat, MY; Dar, AH, Fractional vector-valued nonuniform MRA and associated wavelet packets on \(L^2 (\mathbb{R},\mathbb{C}^M)\), Fract. Calc. Appl. Anal., 25, 2, 687-719, 2022 · Zbl 1503.42029 · doi:10.1007/s13540-022-00035-1 |
| [12] | Bogachev, V.I.: Measure Theory, vols. I. II. Springer, Berlin (2007) · Zbl 1120.28001 |
| [13] | Bratteli, O.; Jorgensen, PET, Endomorphisms of \({\cal{B} }({\cal{H} })\). II. Finitely correlated states on \({\cal{O} }_n\), J. Funct. Anal., 145, 2, 323-373, 1997 · Zbl 0897.46051 · doi:10.1006/jfan.1996.3033 |
| [14] | Bratteli, O., Jorgensen, P.E.T.: A connection between multiresolution wavelet theory of scale \(N\) and representations of the Cuntz algebra \(\cal{O}_N\). In: Operator Algebras and Quantum Field Theory (Rome, 1996), pp. 151-163. Int. Press, Cambridge (1997) · Zbl 0915.46048 |
| [15] | Bratteli, O.; Jorgensen, PET, Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale \(N\), Integr. Equ. Oper. Theory, 28, 4, 382-443, 1997 · Zbl 0897.46054 · doi:10.1007/BF01309155 |
| [16] | Bratteli, O., Jorgensen, P.E.T.: Iterated function systems and permutation representations of the Cuntz algebra. Mem. Am. Math. Soc. 139(663), x+89 (1999) · Zbl 0935.46057 |
| [17] | Bruin, H.; Hawkins, J., Rigidity of smooth one-sided Bernoulli endomorphisms, N. Y. J. Math., 15, 451-483, 2009 · Zbl 1189.37002 |
| [18] | Bruin, H.: Topological and ergodic theory of symbolic dynamics, Graduate Studies in Mathematics, vol. 228. American Mathematical Society, Providence (2022) · Zbl 1514.37002 |
| [19] | Christoffersen, NJ; Dutkay, DE, Representations of Cuntz algebras associated to random walks on graphs, J. Oper. Theory, 88, 1, 139-170, 2022 |
| [20] | Cornfeld, I.P., Fomin, S.V., Sinaĭ, Y.G.: Ergodic theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245. Springer, New York (1982). Translated from the Russian by A. B. Sosinskiĭ · Zbl 0493.28007 |
| [21] | Cuntz, J., Simple \(C^*\)-algebras generated by isometries, Commun. Math. Phys., 57, 2, 173-185, 1977 · Zbl 0399.46045 · doi:10.1007/BF01625776 |
| [22] | Dajani, KG; Hawkins, JM, Examples of natural extensions of nonsingular endomorphisms, Proc. Am. Math. Soc., 120, 4, 1211-1217, 1994 · Zbl 0820.28008 · doi:10.1090/S0002-9939-1994-1174489-4 |
| [23] | Dougherty, R.; Jackson, S.; Kechris, AS, The structure of hyperfinite Borel equivalence relations, Trans. Am. Math. Soc., 341, 1, 193-225, 1994 · Zbl 0803.28009 · doi:10.1090/S0002-9947-1994-1149121-0 |
| [24] | Dutkay, DE; Jorgensen, PET, Hilbert spaces built on a similarity and on dynamical renormalization, J. Math. Phys., 47, 5, 2006 · Zbl 1117.46017 · doi:10.1063/1.2196750 |
| [25] | Dutkay, DE; Jorgensen, PET, Martingales, endomorphisms, and covariant systems of operators in Hilbert space, J. Oper. Theory, 58, 2, 269-310, 2007 · Zbl 1134.47305 |
| [26] | Dutkay, DE; Jorgensen, PET, Monic representations of the Cuntz algebra and Markov measures, J. Funct. Anal., 267, 4, 1011-1034, 2014 · Zbl 1303.47030 · doi:10.1016/j.jfa.2014.05.016 |
| [27] | Dutkay, D.E., Jorgensen, P.E.T.: The role of transfer operators and shifts in the study of fractals: encoding-models, analysis and geometry, commutative and non-commutative. In: Geometry and Analysis of Fractals, Springer Proc. Math. Stat., vol. 88, pp. 65-95. Springer, Heidelberg (2014) · Zbl 1318.28022 |
| [28] | Dutkay, DE; Jorgensen, PET, Representations of Cuntz algebras associated to quasi-stationary Markov measures, Ergod. Theory Dyn. Syst., 35, 7, 2080-2093, 2015 · Zbl 1352.37027 · doi:10.1017/etds.2014.37 |
| [29] | Dutkay, DE; Jorgensen, PET; Silvestrov, S., Decomposition of wavelet representations and Martin boundaries, J. Funct. Anal., 262, 3, 1043-1061, 2012 · Zbl 1242.28025 · doi:10.1016/j.jfa.2011.10.010 |
| [30] | Eisner, T., Farkas, B., Haase, M., Nagel, R.: Operator theoretic aspects of ergodic theory. Graduate Texts in Mathematics, vol. 272. Springer, Cham (2015) · Zbl 1353.37002 |
| [31] | Fabec, RC, Induced group actions, representations and fibered skew product extensions, Trans. Am. Math. Soc., 301, 2, 489-513, 1987 · Zbl 0623.28013 · doi:10.1090/S0002-9947-1987-0882701-2 |
| [32] | Fabec, R.C.: Fundamentals of infinite dimensional representation theory. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 114. Chapman & Hall/CRC, Boca Raton (2000) · Zbl 0964.22001 |
| [33] | Feng, D-J; Simon, K., Dimension estimates for \(C^1\) iterated function systems and repellers. Part II, Ergod. Theory Dyn. Syst., 42, 11, 3357-3392, 2022 · Zbl 1525.28003 · doi:10.1017/etds.2021.92 |
| [34] | Hawkins, JM, Amenable relations for endomorphisms, Trans. Am. Math. Soc., 343, 1, 169-191, 1994 · Zbl 0828.28005 · doi:10.1090/S0002-9947-1994-1179396-3 |
| [35] | Hawkins, J.: Ergodic dynamics—from basic theory to applications, Graduate Texts in Mathematics, vol. 289. Springer, Cham (2021) · Zbl 1509.37001 |
| [36] | Hawkins, JM; Silva, CE, Noninvertible transformations admitting no absolutely continuous \(\sigma \)-finite invariant measure, Proc. Am. Math. Soc., 111, 2, 455-463, 1991 · Zbl 0729.28012 |
| [37] | Jorgensen, PET; Kornelson, K.; Shuman, K., Harmonic analysis of iterated function systems with overlap, J. Math. Phys., 48, 8, 2007 · Zbl 1152.81497 · doi:10.1063/1.2767004 |
| [38] | Jorgensen, P.E.T., Kornelson, K.A., Shuman, K.L.: Iterated function systems, moments, and transformations of infinite matrices. Mem. Am. Math. Soc. 213(1003), x+105 (2011) |
| [39] | Jorgensen, PET, A duality for endomorphisms of von Neumann algebras, J. Math. Phys., 37, 3, 1521-1538, 1996 · Zbl 0897.46052 · doi:10.1063/1.531447 |
| [40] | Jorgensen, P.E.T.: Ruelle operators: functions which are harmonic with respect to a transfer operator. Mem. Am. Math. Soc. 152(720), viii+60 (2001) · Zbl 0995.46046 |
| [41] | Jorgensen, P.E.T.: Analysis and probability: wavelets, signals, fractals, Graduate Texts in Mathematics, vol. 234. Springer, New York (2006) · Zbl 1104.42001 |
| [42] | Jorgensen, P.E.T.: Harmonic analysis, CBMS Regional Conference Series in Mathematics, vol. 128. American Mathematical Society, Providence (2018). Smooth and non-smooth, Published for the Conference Board of the Mathematical Sciences · Zbl 1432.42023 |
| [43] | Jorgensen, PET; Paolucci, AM, States on the Cuntz algebras and \(p\)-adic random walks, J. Aust. Math. Soc., 90, 2, 197-211, 2011 · Zbl 1250.47039 · doi:10.1017/S1446788711001212 |
| [44] | Jorgensen, PET; Song, M-S, Markov chains and generalized wavelet multiresolutions, J. Anal., 26, 2, 259-283, 2018 · Zbl 1402.60094 · doi:10.1007/s41478-018-0139-9 |
| [45] | Jorgensen, P.; Tian, F., Transfer operators, induced probability spaces, and random walk models, Markov Process. Relat. Fields, 23, 2, 187-210, 2017 · Zbl 1500.47123 |
| [46] | Jorgensen, P.; Tian, F., Dynamical properties of endomorphisms, multiresolutions, similarity and orthogonality relations, Discrete Contin. Dyn. Syst. Ser. S, 12, 8, 2307-2348, 2019 · Zbl 1447.81147 |
| [47] | Jorgensen, P.; Tian, J., Noncommutative boundaries arising in dynamics and representations of the Cuntz relations, Numer. Funct. Anal. Optim., 41, 5, 571-620, 2020 · Zbl 1443.47086 · doi:10.1080/01630563.2019.1665544 |
| [48] | Kechris, A.S.: Classical descriptive set theory. Graduate Texts in Mathematics, vol. 156. Springer, New York (1995) · Zbl 0819.04002 |
| [49] | Medhi, R.; Viswanathan, P., On the code space and Hutchinson measure for countable iterated function system consisting of cyclic \(\phi \)-contractions, Chaos Solitons Fractals, 167, 2023 · doi:10.1016/j.chaos.2022.113011 |
| [50] | Picklo, MJ; Ryan, JK, Enhanced multiresolution analysis for multidimensional data utilizing line filtering techniques, SIAM J. Sci. Comput., 44, 4, A2628-A2650, 2022 · Zbl 1497.65179 · doi:10.1137/21M144013X |
| [51] | Przytycki, F., Urbański, M.: Conformal fractals: ergodic theory methods. London Mathematical Society Lecture Note Series, vol. 371. Cambridge University Press, Cambridge (2010) · Zbl 1202.37001 |
| [52] | Rohlin, V.A.: Selected topics from the metric theory of dynamical systems. Uspehi Matem. Nauk (N.S.) 2(30), 57-128 (1949) · Zbl 0032.28403 |
| [53] | Rohlin, V.A.: On the fundamental ideas of measure theory. Mat. Sbornik N.S. 25(67), 107-150 (1949) · Zbl 0033.16904 |
| [54] | Rohlin, VA, Exact endomorphisms of a Lebesgue space, Izv. Akad. Nauk SSSR Ser. Mat., 25, 499-530, 1961 · Zbl 0107.33002 |
| [55] | Roychowdhury, L.; Roychowdhury, MK, Quantization for a probability distribution generated by an infinite iterated function system, Commun. Korean Math. Soc., 37, 3, 765-800, 2022 · Zbl 1505.60026 |
| [56] | Silva, CE, On \(\mu \)-recurrent nonsingular endomorphisms, Isr. J. Math., 61, 1, 1-13, 1988 · Zbl 0645.28010 · doi:10.1007/BF02776298 |
| [57] | Simmons, D., Conditional measures and conditional expectation; Rohlin’s disintegration theorem, Discrete Contin. Dyn. Syst., 32, 7, 2565-2582, 2012 · Zbl 1244.28002 · doi:10.3934/dcds.2012.32.2565 |
| [58] | Urbański, M., Roy, M., Munday, S.: Non-invertible dynamical systems. Vol. 1. Ergodic theory—finite and infinite, thermodynamic formalism, symbolic dynamics and distance expanding maps, De Gruyter Expositions in Mathematics, vol. 69. De Gruyter, Berlin (2022) · Zbl 1493.37003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
